Question: $\overline{AB} = 52$ $\overline{AC} = {?}$ $A$ $C$ $B$ $52$ $?$ $ \sin( \angle BAC ) = \dfrac{5}{13}, \cos( \angle BAC ) = \dfrac{12}{13}, \tan( \angle BAC ) = \dfrac{5}{12}$
Explanation: $\overline{AB}$ is the hypotenuse $\overline{AC}$ is adjacent to $\angle BAC$ SOH CAH TOA We know the hypotenuse and need to solve for the adjacent side so we can use the cos function (CAH) $ \cos( \angle BAC ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\overline{AC}}{\overline{AB}}= \frac{\overline{AC}}{52} $ Since we have already been given $\cos( \angle BAC )$ , we can set up a proportion to find $\overline{AC}$ $ \cos( \angle BAC ) = \dfrac{12}{13} = \frac{\overline{AC}}{52}$ Simplify. $\overline{AC} = 48$